In this section, we prove thm:main for general random vectors \(X \in \mathbb {R}^{m}\)  satisfying \(\Vert X\Vert _{2} \le 1\)  almost surely. As in [1]}, we use principal component analysis to reduce to the case where \(m = c\log {k}\) , for a sufficiently small absolute constant \(c > 0\) . However, our treatment of the dimension-reduced problem is rather different from [1]}. Indeed, whereas [1]} partitions the dimension-reduced random vector according to the closest point in a volumetric epsilon-net (thereby, only exploiting the information that \(\Vert X\Vert _{2} \le 1\)  almost surely), our clustering scheme also takes into account the distributional profile of the dimension-reduced random vector; briefly, we place each `heavy' point into its own cluster, place nearby points, which are `collectively light' into a single cluster, and for the intermediate case, adopt a randomized rounding scheme to cluster the points. In particular, our proof provides another instance where nets based on randomized rounding provide better control than volumetric nets (see [4]}, [5]}, [6]} for some other recent examples).
